Editorial

Can you replicate a call option with two European digital options?

A European digital pays one unit of currency (domestic or foreign) at a predefined expiry date, if on that date the observed FX spot fixes above a predefined strike level. Can you statically replicate a call option with two European digital options?

The question may seem rather elementary but, asking it at quant candidates during technical interviews, I was surprised by the small number of correct answers. In fact, most readers will probably be more familiar with the inverse question: “can you statically replicate a European digital option with call options?”

The latter can be answered by taking the limit of a call spread when the two strikes are converging to each other (and to the strike of the European digital). This is a typical trader’s approach to the risk-management of digitals.

Coming back to our original question, suppose that you own a EURGBP call option with a maturity of 1Y and a strike of 0.85 (GPB per EUR). Now let me simply describe the cash flows at maturity in plain English:

At maturity, if the EURGBP spot fixes above 0.85, you exercise your call option:

  • You pay 0.85 (=strike K) GBP.
  • You receive 1 EUR.

These two legs correspond respectively to

  • A SHORT domestic European digital with strike K and notional K
  • A LONG foreign European digital with strike K and notional 1.

The call option is thus exactly replicated by the difference between two digitals, one paying K units of domestic currency (=cash-or-nothing), and one paying 1 unit of foreign currency (=asset-or-nothing), if the spot fixes above the strike.

Describing the call payoff in mathematical terms, the decomposition is straightforward:

Newsletter November 2019 Can you replicate a call option with two European digital options? - MathFinance

The factor S in the first term comes from the conversion of the foreign digital payoff (in EUR) into domestic currency GBP.  Equation (1) is graphically represented in Figure 1.

Newsletter November 2019 Can you replicate a call option with two European digital options? - MathFinance

 

Figure 1: geometric decomposition of the call payoff into a domestic and a foreign digital.

Note that so far, we haven’t made any model assumption. This is a static replication which holds just by payoff construction. Let us stress that this replication also holds for the Greeks, not only for the price.

Expressing the (undiscounted) call price under the Black model, one immediately recognizes the same replication pattern:

Newsletter November 2019 Can you replicate a call option with two European digital options? - MathFinance

Where N(d1) and N(d2) represent respectively the probability of being “In-The-Money” (S>K) at time T under a foreign (EUR) and a domestic (GBP) risk-neutral measure.

As an independent verification, Figure 2 presents results obtained with SuperDerivatives, pricing our EURGBP example. We put together a small portfolio containing a long position in the EURGBP Call (column1), and a short position in its static replication with Digitals (columns 2 and 3), so that the net total position should be exactly zero.

Newsletter November 2019 Can you replicate a call option with two European digital options? - MathFinance

 

Figure 2: pricing a (Long) EURGBP call and its (Short) static replication in SuperDerivatives.

 As expected, the (mid) price of the portfolio is almost exactly zero (to some minimal numerical noise). This is also the case for the Greeks, except for Delta where we observe a small – yet material- difference of 3% between the call (D=59%) and its static replication (D =56%). This example shows that, despite the apparent simplicity of the approach, it is relevant as soon as you deviate from the plain constant-vol Black-Scholes mode. In the present example, the small Delta difference could come from slight numerical differences in the way the volatility smile is incorporated in foreign and domestic Digitals.

Whether you are a model developer, a model user or a validator, this kind of replication very often comes in handy, and should be used without moderation for testing purpose, or for extending the scope of products in your library.

Let me finally mention that simple extensions are possible be adding common exotic features to the European Call and Digitals. For example, a Call with American barrier(s) will be similarly replicated with domestic and foreign Digitals sharing the same American barrier(s).

 

Dr. Frédéric Bossens

Director, Senior Quant

 


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